Say I'm given a probability space. I need to define a function that looks at a $\sigma$-algebra and outputs a number.
For example, fix a set $B\subseteq\Omega$. A simple function would be to output 1 if the $\sigma$-algebra contains $B$ and zero otherwise.
I am interested in other (more complicated) functions, but I was shaky when trying to write formally the definitions. Could somebody provide a reference where they deal with objects like this?
I am ultimately interested in something more, which is a function of a filtration, but I guess the jump is easy once I fully understand the previous.
Cheers